A366388 - OEIS

%I #21 Oct 23 2023 15:07:39

%S 0,0,1,0,2,1,1,0,2,2,3,1,2,1,3,0,2,2,1,2,2,3,3,1,4,2,3,1,3,3,4,0,4,2,

%T 3,2,2,1,3,2,3,2,2,3,4,3,4,1,2,4,3,2,1,3,5,1,2,3,3,3,3,4,3,0,4,4,2,2,

%U 4,3,3,2,3,2,5,1,4,3,4,2,4,3,4,2,4,2,4,3,2,4,3,3,5,4,3,1,5,2,5,4,3,3,4,2,4

%N The number of edges minus the number of leafs in the rooted tree with Matula-Goebel number n.

%C Number of iterations of A366385 needed to reach the nearest power of 2.

%H Antti Karttunen, <a href="/A366388/b366388.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>

%F Totally additive with a(2) = 0, and for n > 1, a(prime(n)) = 1 + a(n).

%F a(n) = A196050(n) - A109129(n).

%F a(2n) = a(A000265(n)) = a(n).

%e See illustrations in A061773.

%t Array[-1 + Length@ NestWhileList[PrimePi[#2]*#1/#2 & @@ {#, FactorInteger[#][[-1, 1]]} &, #, ! IntegerQ@ Log2[#] &] &, 105] (* _Michael De Vlieger_, Oct 23 2023 *)

%o (PARI) A366388(n) = if(n<=2, 0, if(isprime(n), 1+A366388(primepi(n)), my(f=factor(n)); (apply(A366388, f[, 1])~ * f[, 2])));

%o (PARI)

%o A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);

%o A366385(n) = { my(gpf=A006530(n)); primepi(gpf)*(n/gpf); };

%o A366388(n) = if(n && !bitand(n,n-1),0,1+A366388(A366385(n)));

%Y Cf. A000720, A006530, A052126, A061395, A061773, A366385.

%Y Cf. A109129 (gives the exponent of the nearest power of 2 reached), A196050 (distance to the farthest power of 2, which is 1).

%Y Cf. also A329697, A331410.

%K nonn

%O 1,5

%A _Antti Karttunen_, Oct 23 2023